Minimum number of plates

Column length and diameter these are set according to the minimum number of plates required and the maximum acceptable pressure-drop, and calculated by taking into account the constraints given by the Van Deemter and Darcy s laws. 

The conditions of total liquid reflux in a column also represent the minimum number of plates required for a given separation. Under such conditions the column has zero production of product, and infinite heat requirements, and Lj/Vs = 1.0 as shown in Figure 8-15. This is the limiting condition for the number of trays and is a convenient measure of the complexity or difficulty of separation. 

The minimum number of plates [129], for infinite time for separation ... 

Solving the equation by trial and error shows that Xj = 0.18530. Solving for the minimum number of plates required ... 

Calculate minimum number of plates and minimum reflux ratio... 

This shows that for a desired degree of resolution three conditions have to be met (a) the peaks have to be retained on the column (k 2 > 0), (b) the peaks have to be separated from each other (a > 1), and (c) the column must develop some minimum number of plates. 

The Underwood and Fenske equations may be used to find the minimum number of plates and the minimum reflux ratio for a binary system. For a multicomponent system nm may be found by using the two key components in place of the binary system and the relative volatility between those components in equation 11.56 enables the minimum reflux ratio Rm to be found. Using the feed and top compositions of component A ... 

Fenske s equation may be used to find the minimum number of plates. 

Cp = cost per square foot of plate area, /ft2 A = column cross-sectional area, ft2 N = number of plates A min = minimum number of plates Cs = cost of shell, /ft3 H = distance between plates, ft Cf = cost of feed pump, ... 

Seader and Henley (1998) considered the separation of a ternary mixture in a batch distillation column with B0 = 100 moles, xB0 = = <0.33, 0.33, 0.34> molefraction, relative volatility a= <2.0, 1.5, 1.0>, theoretical plates N = 3, reflux ratio R = 10 and vapour boilup ratio V = 110 kmol/hr. The column operation was simulated using the short-cut model of Sundaram and Evans (1993a). The results in terms of reboiler holdup (Bj), reboiler composition profile (xBI), accumulated distillate composition profile (xa), minimum number of plates (Nmin) and minimum... 

Note that while recoveries and product purity specifications for the key components are selected as independent decision variables in the outer loop, these should not be assigned arbitrary values. It is possible that certain purities may not be achievable with the current column configuration (this can be checked against the minimum number of plates, Nmin required for the given separation, Chapter 3), or due to the presence of azeotropes, etc. At the very least, the specifications must be consistent with the amounts of the various species present in the initial charge and in the feed state to each separation task (i.e. the overall mass balance must be satisfied). Thus care must be taken in selecting the outer loop specifications and bounds. 

Diwekar (1995) also noted a case where the performance of CBD column was better than that of a continuous column. Although there was no mention of the location of the feed plate, it is possible that the performance was not only affected by the location of the feed plate but also by the small number of plates (only six plates). In fact, Abrams et al. (1987) suggested that the comparison of a CBD operation with that of a continuous column is worth making only when the actual number of plates in both columns is at least twice the minimum. In case of Diwekar (1995) the minimum number of plates for continuous column was 6.72 (greater than the actual number of plates). See the original references for further details. 

Although the goal of achieving the separation with a minimum number of plates appears... 

Maximization of the minimum resolution value observed in the chromatogram (eqn.4.25) corresponds to aiming at the minimum number of plates required to effectuate the separation. 

The flow rates given previously lead to 100% purities in the case of an ideal TMB, equivalent to an infinite number of plates. The approach used here is to keep these flow rates and to seek the minimum number of plates Nm required to reach the required purities as high as 99%. 

In a first approximation, the previous complex calculation can be avoided by assuming that the minimum number of plates required to obtain adequate purities is identical to the number of plates that was required for the TMB. 

In making estimates of the minimum number of plates required to achieve a desired level of separation it is important to realize that the often reproduced chart by Glueckauf is misleading. It is applicable with negligible error only when k is so large as to be of little chromatographic interest. 

Figure 4.10 McCabe-Thiele analysis for total reflux and minimum number of plates.

As the specified value of the reflux ratio (Ll/D) is decreased, the intersection of the two operating lines moves closer to the equilibrium curve and the minimum number of plates required to effect the specified separation (xB = 0.05, XD = 0.96) increases. On the other hand, as L /D is decreased, the condenser and reboiler duties decrease. The minimum reflux ratio is the smallest one which can be used to effect the specified separation. This reflux ratio requires infinitely many plates in each section as demonstrated in Fig. 1-10. It should be noted that for this case, the plates at and adjacent to the feed plate have the same composition. (In the case of multicomponent systems, these limiting conditions do not necessarily occur at and adjacent to the feed plate as discussed in Chap. 11). From the standpoint of construction costs, this reflux ratio is unacceptable because infinitely many plates are required, which demands a column of infinite height. 

The upper and lower bounds on D and W are established by use of the results obtained by finding the solutions at total reflux at the maximum and minimum number of plates as described in App. 9-1. The lower bounds on D and W were taken to be a fraction of the smaller values found at total reflux. Similarly, the upper bounds on D and W were taken to be a fraction larger than their corresponding values at total reflux. For the examples presented, the multipliers of 0.9 and 1.1 were used to compute the lower and upper bounds, respectively. 

The desirable characteristics of the initial search, which was originally proposed by Srygley and Holland19 for minimizing the number of equilibrium stages at a fixed reflux ratio, are clearly demonstrated in the solutions of the examples considered. The initial search produced values for the minimum number of plates which wefe consistently close to the solution values obtained by the final search of procedure 1. 

In designing a plant. Go and P might first be set. At several values of F, Eqs. (13.151) through (13.154) would then be used to evaluate G, Lg, G/, and Z,/,. The ratios a.gLjGg and Gh/LhOif, would be determined the optimum value of F that leads to the minimum number of plates is the one at which... 

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